No, because conjured objects don’t last.
Conjuring birds using Avis doesn’t conflict with food being one of the five exceptions to Gamp’s Law of Elemental Transfiguration, because things that are conjured from thin air won’t last.
Q: It seems that the wizards and witches at Hogwarts are able to conjure up many things, such as food for the feasts, chairs and sleeping bags. . .if this is so, why does the wizarding world need money? What are the limitations on the material objects you can conjure up? It seems unnecessary that the Weasleys would be in such need of money. . . (Jan Campbell)
A: Very good question (well done, Jan!!). There is legislation about what you can conjure and what you can't. Something that you conjure out of thin air will not last. This is a rule I set down for myself early on. I love these logical questions!
- South West News Service Interview with J.K. Rowling (July 8th, 2000)
Therefore, any attempt to eat the birds conjured by Avis would be futile, because they’ll soon vanish. Even if they don’t vanish before they can be killed, they will eventually, so they wouldn’t provide any lasting nutrition even if they can be killed and cooked.